A positive definite kernel \(\kappa \)  is said to be consistent if every strong Cauchy net \((\mu _s)\subset \mathcal {E}^+\)  converges strongly to any of its vague limit points \(\mu \)  [1]}.Since the space \(\mathfrak {M}\)  equipped with the vague topology does not necessarily satisfy the first axiom of countability, the vague convergence cannot in general be described in terms of sequences. We follow Moore and Smith's theory of convergence
[2]}, based on the concept of nets. However, if a locally compact space \(X\)  is second-countable, then the space \(\mathfrak {M}\)  is first-countable [3]}, and the use of nets in \(\mathfrak {M}\)  may often be avoided. (Such \(\mu \)  do exist e.g. if the kernel \(\kappa \)  is strictly positive definite [1]}.)
